Four-loop relation between the $\bar{\rm MS}$ and on-shell quark mass

0.11 MB·

by  Peter Marquard

Four-loop relation between the MS and on-shell
6 quark mass
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PeterMarquard
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DeutschesElektronenSynchrotronDESY,Platanenallee6,15738Zeuthen,Germany

] E-mail: [email protected]
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AlexanderV.Smirnov
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p ScientificResearchComputingCenter,MoscowStateUniversity,119991,Moscow,Russia
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E-mail: [email protected]
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VladimirA. Smirnov

1 SkobeltsynInstituteofNuclearPhysics,MoscowStateUniversity,119991,Moscow,Russia
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E-mail: [email protected]
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MatthiasSteinhauser∗
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3 InstitutfürTheoretischeTeilchenphysik,KarlsruheInstituteofTechnology(KIT),76128
0 Karlsruhe,Germany
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1 E-mail: [email protected]
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1 Inthiscontributionwe discussthefour-looprelationbetweentheon-shellandMSdefinitionof
:
v heavyquarkmasseswhichisappliedtothetop,bottomandcharmcase. Wealsopresentrelations
i
X betweentheMSquarkmassandvariousthresholdmassdefinitionsanddiscusstheuncertaintyat
r next-to-next-to-next-to-leadingorder.
a
12thInternationalSymposiumonRadiativeCorrections(Radcor2015)andLoopFestXIV(Radiative
CorrectionsfortheLHCandFutureColliders)
15-19June,2015
UCLADepartmentofPhysics&AstronomyLosAngeles,USA
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
1. Introduction
In the Standard Model (and many of its extensions) quark masses enter as fundamental pa-
rameters m into the underlying Lagrange density. Once quantum corrections are considered one
q
has to fix the precise definition of m . For heavy quarks a natural one is the on-shell definition
q
whereonerequiresthatthequarkpropagatorS (q)hasapoleforq2=(MOS)2. However,thereare
q q
manysituationswhereotherdefinitionsaremoreconvenient. Asanexamplewementionthedecay
rate of Higgs bosons to bottom quarks where, when expressed in terms of the MS bottom quark
massevaluatedattheappropriate scale,potentially largelogarithmsareautomatically summedup.
A further example isthe threshold production of top quark pairs in electron-positron annihilation.
For this process one has to adopt a properly constructed (so-called) threshold mass which, on the
one hand, isof short-distance nature as theMSmass. Ontheother hand, ithassimilar features as
theon-shell mass. Inparticular, ithasaphysicaldefinitionatthreshold.
It is important to have precise relations among the various mass definitions. In Ref. [1] four-
loopcorrections totherelationbetweentheMSandon-shellheavyquarkdefinitionhasbeencom-
puted. This result has been used to derive next-to-next-to-next-to-leading order (N3LO) relations
among theMS and themost popular threshold masses, namely the PS[2], 1S [3, 4, 5]and RS[6]
masses.
The relation between the MS (m) and the on-shell mass (M) is obtained by considering in a
firststeptheirrelation tothebaremass,m :
0
m0=ZMSm, m0=ZOSM. (1.1)
m m
Here,ZMS isknowntofive-looporder[7]. However,inourcalculation, onlythefour-loop resultis
m
necessary[8,9,10]. ZOSiscomputedfromthescalarandvectorcontributionofthequarktwo-point
m
function withon-shellexternalmomentumvia
ZOS = 1+S (q2=M2)+S (q2=M2). (1.2)
m V S
One-,two-andthree-loopresultstoZOShavebeencomputedinRefs.[11],[12]and[13,14,15,16],
m
respectively. Four-loopresulthaverecently beencomputedinRef.[1].
Byconstruction, theratioofthetwoequations in(1.1)isfinitewhichleadsto
m(m )
z (m ) = . (1.3)
m
M
Itisconvenient tocasttheperturbative expansion intheform
z (m ) = (cid:229) a s nz(n), (1.4)
m (cid:16)p (cid:17) m
n≥0
(0)
with z =1. In the next section we present results for z up to four-loop order and discuss the
m m
numerical effects for charm, bottom and top quarks. Afterwards we consider in Section 3 the
relation betweentheMSandvarious threshold masses. Section4containsourconclusions.
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Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
2. Four-loopMS-on-shell relation
For the computation of the fermion self energy we use an automated setup which generates
all contributing amplitudes with the help of qgraf [17]. The output is transformed to FORM3-
readable [18] input using q2e and exp [19, 20]. Afterwards projectors for the scalar and vector
partareapplied,tracesaretakenandthescalarproductsinthenumeratoraredecomposedinpropa-
gatorfactors. Thisleadstoseveralmilliondifferentintegralsencodedinfunctionswith14different
indices whichbelongto100different integralfamilies.
The Laporta algorithm [21] is applied to each family using FIRE5 [22] and crusher [23]
which are written in C++. Then we use the code tsort [24], which is part of the latest FIRE
version, to reveal relations between primary master integrals (following recipes of Ref. [25]) and
endupwith386four-loop massiveon-shellpropagator integrals, i.e. with p2=M2.
Up to this point the whole calculation is analytic. However, at the moment not all master
integrals could be evaluated analytically but only numerically using FIESTA [26, 27, 28] which
leadstoanaccuracyofaboutfivetosixdigitsforthehigheste expansion term. Forsomeintegrals
a two- or threefold Mellin Barnes representation could be derived which enabled us to obtain a
precision ofmorethaneight,insomecasesevenmorethan20digits.
Foreachintegralwhichisevaluatednumerically,eache coefficientgetsaseparateuncertainty
assigned. SinceitresultsfromanumericalMonteCarlointegrationweinterpretitasastandardde-
viationandcombinetheindividualuncertaintiesinthefinalexpressionquadratically. Furthermore,
wemultiplytheuncertainty inthefinalresultfortheMSandon-shellrelation byafactorfive.
Notethatwehaveperformedthecalculationallowingforageneralgaugeparameterx keeping
terms up to order x 2 in the expression wegive to the reduction routines. We have checked that x
dropsoutaftermassrenormalization butbeforeinserting themasterintegrals.
In the following we show the MS-on-shell relation in the form where the on-shell mass is
computed from the MS mass. We discuss the top, bottom and charm quark case and use as input
the following MS masses: m ≡ m (m ) = 163.643 GeV, m ≡ m (m ) = 4.163 GeV [29], and
t t t b b b
m (3 GeV) = 0.986 GeV [29]. The corresponding values for the strong coupling are given by
c
a (6)(m )=0.1088,a (5)(m )=0.2268,anda (4)(3GeV)=0.2560. Theyhavebeencomputedfrom
s t s b s
a (5)(M )=0.1185 [30] using RunDec[31, 32]. In the case of the charm quark we also provide
s Z
resultsform =m usingtheinputvaluesm ≡m (m )=1.279GeVanda (4)(m )=0.3923. Note
c c c c s c
that the choice m =3GeVispreferable since ithasthe advantage thatlow renormalization scales
m ≈m areavoided. Ourresultsread
c
M = m 1+0.4244a +0.8345a 2+2.375a 3+(8.49±0.25)a 4
t t s s s s
(cid:0) (cid:1)
= 163.643+7.557+1.617+0.501+0.195±0.005 GeV, (2.1)
M = m 1+0.4244a +0.9401a 2+3.045a 3+(12.57±0.38)a 4
b b(cid:0) s s s s(cid:1)
= 4.163+0.401+0.201+0.148+0.138±0.004 GeV, (2.2)
M = m (3GeV) 1+1.133a +3.119a 2+10.98a 3+(51.29±0.52)a 4
c c s s s s
(cid:0) (cid:1)
= 0.986+0.286+0.202+0.182+0.217±0.002 GeV, (2.3)
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Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
M = m 1+0.4244a +1.0456a 2+3.757a 3+(17.36±0.52)a 4
c c(cid:0) s s s s(cid:1)
= 1.279+0.213+0.206+0.290+0.526±0.016 GeV. (2.4)
Forthetopquarkthehigherordercorrections becomesuccessivelysmallerbyafactortwotothree
leading to a four-loop correction term of about 200 MeV. This is the same order of magnitude as
theintrinsic uncertainty oftheMS-on-shell relation givenbyL . Thefour-loop corrections are
QCD
stillsmallerthanthecurrentuncertainty ofthetopquarkformtheTEVATRONandtheLHC[33].
However,theyarenotnegligible.
Forthebottom andcharm quarkcase thesituation iscompletely different. Noconvergence is
observed when increasing the loop order. In the case of the charm quark where m (m ) is chosen
c c
as a starting point one even observes a four-loop coefficient which is almost twice as large as the
three-loop one.
From the above results one can conclude that the immediate application of the MS-on-shell
relation isonlymeaningful forthetopquarkcase. Forthelighterquarks theon-shell massparam-
eter should beavoided. Ifnecessary anappropriately chosen threshold massshould beused aswe
willdiscussinthenextsection.
In the following we present for the top quark mass the inverted relation of Eq. (2.1) which
reads1
m = M 1−0.4244a −0.65441a 2−1.944a 3−(7.23±0.22)a 4
t t s s s s
(cid:0) (cid:1)
= 173.34−7.948−1.324−0.425−0.171±0.005 GeV, (2.5)
whereM =173.34 GeV[33]anda (6)(M)=0.1080 hasbeenused. Thisequation canbeusedto
t s t
computem (m )foragivenvaluefortheon-shellmassM.
t t t
3. Relationbetween MSand threshold massesto N3LO
InthissectionwepresentnumericalresultsfortheMSquarkmassesusinginputvaluesforthe
PS,1SandRSthresholdmasses. Inpracticalapplicationsthelatterareextractedfromcomparisons
withexperimentaldata. ThederivationoftheN3LOrelationsisdiscussedinRef.[1]followingthe
prescriptions providedintheoriginalreferences [2,3,4,5,6].
Table1showsresultsfortheMStopquarkmasscomputed fromthePS,1SandRSthreshold
massvaluesgiveninthefirstandsecondrow. Notethatthesevaluesarechoseninsuchawaythatin
allthreecasesthesameMSmassisobtainedafterapplyingfour-loopcorrections, whichfacilitates
thecomparison. Notealso,thatincontrasttothecorresponding tableinRef.[1]wechooseforthe
factorization scale of thePSmass m =80 GeVinstead of m =20GeV.Thisissuggested bythe
f f
N3LO threshold analysis of s (e+e− →tt¯) performed in Ref. [34]. The factorization scale for the
RSmassiskeptatm =20GeV.
f
In all three cases one observes a rapid convergence of the perturbative series. In fact, the
NNLO term amounts to at most 210 MeV (1S mass), and at N3LO at most 20 MeV (RS mass).
After increasing the four-loop MS-on-shell term by 3%, which is the current uncertainty on the
four-loopcoefficientinEq.(1.4),themassvaluesreducesby6MeV.Combiningthesetwosources
1NotethattheMSvalueusedinEq.(2.1)hasbeenobtainedusingEq.(2.5)tothree-loopaccuracy.
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Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
input mPS= m1S= mRS=
#loops 168.204 172.227 171.215
1 164.311 165.045 164.847
2 163.713 163.861 163.853
3 163.625 163.651 163.663
4 163.643 163.643 163.643
4(×1.03) 163.637 163.637 163.637
Table 1: m(m)in GeVcomputedfromthePS,1SandRSquarkmassusingLOtoN3LOaccuracy. The
t t
numbersinthelastlineareobtainedbytakingintoaccounttheuncertaintyofthefour-loopcoefficient,i.e.,
itisincreasedby3%.
166.0
1 loop
2 loops
165.5 PS 3 loops
4 loops
)
V
e
165.0
G
(

)
t 164.5
m
(
t
m
164.0
163.5
100 150 200 250 300
µ (GeV)
Figure 1: MS top quark mass m(m ) computed from the PS mass with LO, NLO, NNLO and N3LO
t t
accuracyasafunctionoftherenormalizationscaleusedintheMS-thresholdmassrelation.
of uncertainties one ends up in a final uncertainty below 20 MeV which is sufficient for a precise
determination ofm atafuturelinearcollider [34].
t
LetusatthispointhaveacloserlooktothePSmass. InTable1therenormalization scalehas
been fixedto m =m . It isalso interesting toconsider different values of m and compute ina first
t
step m (m )whichisthen evolved to m (m )using renormalization group methods. InFigure 1we
t t t
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Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
163.8
PS
1S
)
V RS
e
3 loops
G
4 loops
(
163.7

)
t
m
(
t
m
163.6
100 150 200 250 300
µ (GeV)
Figure 2: MS top quarkmass m(m) computedfrom the PS, 1S and RS mass with NNLO (dashed)and
t t
N3LO(solidline)accuracyasafunctionoftherenormalizationscaleusedintheMS-thresholdmassrelation.
Form =300GeVthelinesfrombottomtotopcorrespondtothePS,1SandRSmass.
plot theresult form (m )computed frommPS =168.204 GeVusing LO,NLO,NNLOand N3LO
t t t
accuracy (from short-dashed to solid lines). Whereas the LO curve shows a variation of several
hundred MeV the N3LO is basically independent of m . Actually, in the considered range from
m /2 to 2m it varies by less than 20 MeV, a number comparable to the difference between the
t t
NNLOandN3LOresultatthecentral scale m =m .
t
ThebehaviouroftheNNLOandN3LOcurveofFigure1ismagnifiedinFigure2(redcurves).
Inaddition thecorresponding resultsareshownforthe1S(green)andRS(blue)mass. Inallthree
cases one observes a significant improvement of the m dependence when going from NNLO to
N3LO.Furthermore, theN3LOcurvesofallthreethreshold massesonlydependmildlyon m .
InTable2results fortheMSbottom quark massareshown. Theyarecomputed fromthePS,
1S and RS masses as given in the first and second row of the table using LO, NLO, NNLO and
N3LOaccuracy. Similartothetopquarkcase,oneobservesarapidconvergence withashiftbelow
10MeVfromthelastperturbativeorder. Avariationofthefour-loopMS-on-shellcoefficientleads
toashiftof4MeVintheMSmass.
InTable3weshowthecorresponding resultsfortheMScharmquarkmass. Eveninthiscase
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Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
input mPS= m1S= mRS=
#loops 4.483 4.670 4.365
1 4.266 4.308 4.210
2 4.191 4.190 4.172
3 4.161 4.154 4.158
4 4.163 4.163 4.163
4(×1.03) 4.159 4.159 4.159
Table2: m (m )inGeVcomputedfromthePS,1SandRSquarkmassusingLOtoN3LOaccuracy. The
b b
numbersinthelastlineareobtainedbytakingintoaccounttheuncertaintyofthefour-loopcoefficient,i.e.,
itisincreasedby3%. ThefactorizationscalesforthePSandRSmassaresetto2GeV.
input mPS= m1S= mRS=
#loops 1.155 1.552 1.044
1 1.078 1.265 1.028
2 1.021 1.119 1.008
3 0.993 1.033 0.991
4 0.986 0.986 0.986
4(×1.03) 0.984 0.984 0.984
Table 3: m (3 GeV) in GeV computedfromthe PS, 1SandRS quarkmassusing LO to N3LO accuracy.
c
Thenumbersinthelastlineareobtainedbytakingintoaccounttheuncertaintyofthefour-loopcoefficient,
i.e.,itisincreasedby3%. ThefactorizationscalesforthePSandRSmassaresetto2GeV.
weobserveareasonable convergence oftheperturbativeseries. ForthePSandRSmasstheN3LO
corrections areevenbelow10MeV.
4. Conclusions
In this contribution we considered the four-loop relation between the MS and on-shell heavy
quark masses and applied it to the top, bottom and charm case. Whereas the perturbative series
converges wellfortopitdoesnotfortheothertwocases. Thissuggests thattheon-shelltopquark
massisareasonablygoodparameterattheorderof100MeVorevenbetter. Forallthreecasesthe
perturbative relation between thethreshold (PS,1S,RS)andtheMSmassesisperturbatively well
behaved. Thus, in case a threshold mass is determined from a physical quantity like a (threshold)
cross section or a bound state energy it can be related to the corresponding MS mass with high
precision.
Acknowledgments
We thank the High Performance Computing Center Stuttgart (HLRS) and the Supercomput-
ing Center of Lomonosov Moscow State University [35] for providing computing time used for
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Four-looprelationbetweentheMSandon-shellquarkmass MatthiasSteinhauser
the numerical computations with FIESTA.P.M. was supported in part by the EU Network HIG-
GSTOOLSPITN-GA-2012-316704. Thisworkwassupported bytheDFGthrough theSFB/TR9
“ComputationalParticlePhysics”. TheworkofV.S.wassupportedbytheAlexandervonHumboldt
Foundation (HumboldtForschungspreis).
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